# set which has outer measure as infinity can be expressed as disjoint union of countable measurable sets

I am studying the 'Real Analysis' by Royden, 4th edition.

In page $41$, Royden quote the following:

If $m^*(E) = \infty$ and measurable , then $E$ can be expressed as the disjoint union of countable collection $\{ E_k \}$ of measurable sets, each of which has finite outer measure.

How to prove the above statement?

This statement is essentially equivalent to the $\sigma$-finiteness of the measure space. Simply take it to be the disjoint union
$$E = \bigcup_{n= -\infty}^{\infty} E \cap [n, n+1)$$
• Why we need $m^*(E) = \infty$? – Idonknow Mar 1 '17 at 15:21
• @Idonknow You don't, but if the outer measure of $E$ is finite; then the disjoint union can be taken to consist of just the single set $E$. – Starfall Mar 1 '17 at 15:22
• do you mean $E =E$? – Idonknow Mar 1 '17 at 15:44